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L. G. KOVÁCS’ WORK ON LIE POWERS

Part of: Representation theory of groups Lie algebras and Lie superalgebras Linear algebraic groups and related topics

Published online by Cambridge University Press:  09 June 2015

MARIANNE JOHNSON
MARIANNE JOHNSON*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK email [email protected]
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Abstract

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From the mid-1990s onwards, the main focus of L. G. Kovács’ research was on Lie powers. This brief survey presents some of the key results on Lie powers obtained by Kovács and his collaborators, and discusses some subsequent developments and applications of this work.

Keywords

Lie powersmodular representationsintegral representations

MSC classification

Primary: 20G05: Representation theory
Secondary: 17B01: Identities, free Lie (super)algebras 20C20: Modular representations and characters 20C10: Integral representations of finite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 1 , February 2017 , pp. 9 - 19
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Brandt, A. J., ‘The free Lie ring and Lie representations of the full linear group’, Trans. Amer. Math. Soc. 56 (1944), 528–536.Google Scholar
Bryant, R. M. and Schocker, M., ‘The decomposition of Lie powers’, Proc. Lond. Math. Soc. (3) 93(1) (2006), 175–196.Google Scholar
Bryant, R. M. and Stöhr, R., ‘Fixed points of automorphisms of free Lie algebras’, Arch. Math. (Basel) 67(4) (1996), 281–289.CrossRefGoogle Scholar
Bryant, R. M. and Stöhr, R., ‘On the module structure of free Lie algebras’, Trans. Amer. Math. Soc. 352(2) (2000), 901–934.CrossRefGoogle Scholar
Bryant, R. M and Stöhr, R., ‘Lie powers in prime degree’, Q. J. Math. 56(4) (2005), 473–489.Google Scholar
Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Free Lie algebras as modules for symmetric groups’, J. Aust. Math. Soc. Ser. A 67(2) (1999), 143–156.CrossRefGoogle Scholar
Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Invariant bases for free Lie rings’, Q. J. Math. 53(1) (2002), 1–17.Google Scholar
Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Lie powers of modules for groups of prime order’, Proc. Lond. Math. Soc. (3) 84(2) (2002), 343–374.CrossRefGoogle Scholar
Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Lie powers of modules for GL(2, p)’, J. Algebra 260(2) (2003), 617–630.Google Scholar
Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Subalgebras of free restricted Lie algebras’, Bull. Aust. Math. Soc. 72(1) (2005), 147–156.CrossRefGoogle Scholar
Donkin, S. and Erdmann, K., ‘Tilting modules, symmetric functions, and the module structure of the free Lie algebra’, J. Algebra 203(1) (1998), 69–90.Google Scholar
Erdmann, K. and Kovács, L. G., ‘Metabelian Lie powers of the natural module for a general linear group’, J. Algebra 352 (2012), 232–267.Google Scholar
Erdmann, K. and Schocker, M., ‘Modular Lie powers and the Solomon descent algebra’, Math. Z. 253(2) (2006), 295–313.Google Scholar
Gupta, C. K., ‘The free centre-by-metabelian groups’, J. Aust. Math. Soc. 16 (1973), 294–299; Collection of articles dedicated to the memory of Hanna Neumann, III.Google Scholar
Johnson, M., ‘Standard tableaux and Klyachko’s theorem on Lie representations’, J. Combin. Theory Ser. A 114(1) (2007), 151–158.CrossRefGoogle Scholar
Johnson, M. and Stöhr, R., ‘Free central extensions of groups and modular Lie powers of relation modules’, Proc. Amer. Math. Soc. 138(11) (2010), 3807–3814.Google Scholar
Klyachko, A. A., ‘Lie elements in the tensor algebra’, Siberian Math. J. 15(6) (1975), 914–920; translated from Sibirsk. Mat. Zh. 15 (6) (1974), 1296–1304.Google Scholar
Kovács, L. G. and Stöhr, R., ‘Module structure of the free Lie ring on three generators’, Arch. Math. (Basel) 73(3) (1999), 182–185.Google Scholar
Kovács, L. G. and Stöhr, R., ‘Lie powers of the natural module for GL(2)’, J. Algebra 229(2) (2000), 435–462.CrossRefGoogle Scholar
Kovács, L. G. and Stöhr, R., ‘On Lie powers of regular modules in characteristic 2’, Rend. Semin. Mat. Univ. Padova 112 (2004), 41–69.Google Scholar
Kovács, L. G. and Stöhr, R., ‘A combinatorial proof of Klyachko’s theorem on Lie representations’, J. Algebraic Combin. 23(3) (2006), 225–230.Google Scholar
Kovács, L. G. and Stöhr, R., ‘Lie powers of relation modules for groups’, J. Algebra 326 (2011), 192–200.CrossRefGoogle Scholar
Kraśkiewicz, W. and Weyman, J., ‘Algebra of coinvariants and the action of a Coxeter element’, Bayreuth. Math. Schr. 63 (2001), 265–284; Preprint, 1987.Google Scholar
Kuz’min, Yu. V., ‘Free center-by-metabelian groups, Lie algebras and D-groups’, Izv. Akad. Nauk SSSR Ser. Mat. 41(1) (1977), 3–33 (in Russian).Google Scholar
Lee, M. P., ‘Integral representations of dihedral groups of order 2p ’, Trans. Amer. Math. Soc. 110 (1964), 213–231.Google Scholar
V. D. Mazurov (ed.), Kourovka Notebook: Unsolved Problems in Group Theory, 11th edn, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1990.Google Scholar
Schocker, M., ‘The descent algebra of the symmetric group’, in: Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry, Fields Institute Communications, 40 (American Mathematical Society, Providence, RI, 2004), 145–161.Google Scholar
Schur, I., ‘Über die rationalen Darstellungen der allgemeinen linearen Gruppe, (1927)’, in: Gesammelte Abhandlungen, III (eds. Brauer, A. and Rohrbach, H.) (Springer, Berlin, 1973), 68–85.Google Scholar
Selick, P. and Wu, J., ‘Some calculations of Lie(n)max for low n ’, J. Pure Appl. Algebra 212(11) (2008), 2570–2580.Google Scholar
Shmelkin, A. L., ‘Wreath products and varieties of groups’, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 149–170 (in Russian).Google Scholar
Stöhr, R., ‘On torsion in free central extensions of some torsion-free groups’, J. Pure Appl. Algebra 46 (1987), 249–289.Google Scholar
Thrall, R. M., ‘On symmetrized Kronecker powers and the structure of the free Lie ring’, Amer. J. Math. 64 (1942), 371–388.Google Scholar
Wever, F., ‘Über Invarianten in Lie’schen Ringen’, Math. Ann. 120 (1949), 563–580.Google Scholar
Witt, E., ‘Die Unterringe der freien Lieschen Ringe’, Math. Z. 64 (1956), 195–216.Google Scholar